(slides)

On the classification of

modules over elliptic algebras

Koen De Naeghel, talk s´eminaire d’alg`ebre Institut Henri Poincar´e, Paris

Based on joint work with Michel Van den Bergh. Part of this research is unfinished.

1. Motivation

2. Some preliminaries on noncommutative planes

3. Moduli spaces for gk-2 modules

4. Hilbert series of gk-2 modules

1. Motivation

• First Weyl algebra

A₁ = C_{hx, yi/(yx − xy − 1)}

Cannings and Holland, Wilson:

R = { right A₁-ideals}/₌∼ ←→ a n

Cn

where

C_n={X, Y ∈ M_n(C_{)| rk(Y X−XY −id) = 1}/Gln(}C₎

is the n-th Calogero-Moser space

• Berest and Wilson gave a new proof using noncommutative algebraic geometry:

right A₁-ideals A₁ = H/(z − 1)

H = C_{hx, y, zi}

reflexive graded right

certain objects cohP2_{q = H-modules}

H-modules of rank one

up to finite length where yx − xy = z2

zx = xz, zy = yz

Due to a Theorem of Bondal: Db(coh P2_q) RHom P2 q(E,–) −→ ←− – ⊗L_∆E Db(mod ∆)

where E = O(2) ⊕ O(1) ⊕ O and ∆ is the quiver

X₀

−→ −→X1

0 −→Y0 1 −→Y1 2

Z₀

−→ −→Z1

with the relations of ∆ reflecting the defining relations of H:      X₁Z₀ = Z₁X₀ Y₁Z₀ = Z₁Y₀ X₁Y₀ − Y₁X₀ = Z₁Z₀

Under this equivalence:

right A₁-ideal ←→ M ∈ mod ∆ for which

dimM = (n, n, n − 1) and

Hom_∆(M, p) = Hom_∆(p, M ) = 0 ∀p ∈ P1

which is equivalent with

dimM = (n, n, n − 1) and M (Z₀), M (Z₁) are surjective Using the relations of ∆:

M (X₀)M (Z₀)−1, M (Y₀)M (Z₀)−1 !

• There are more algebras inducing a P2_q

Interesting class:

A = Artin-Schelter regular algebra in three variables

are determined by geometric data (E, σ, L). Set

R = { reflexive graded right

A-modules of rank one }/_=,∼ _sh

If E smooth and O(σ) = ∞:

M ∈ R ←→ M ∈ mod ∆ for which

dimM = (n, n, n − 1) and

Hom_∆(M, p) = Hom_∆(p, M ) = 0 ∀p ∈ E More subtle to handle!

In case A is a Sklyanin algebra i.e.

Skl₃(a, b, c) = khx, y, zi/(f₁, f₂, f₃)

where k algebraically closed field char. zero and      f₁ = ayz + bzy + cx2 f₂ = azx + bxz + cy2 f₃ = axy + byx + cz2 Then R ←→ a n D_n

D_n smooth affine connected variety of dim 2n D₀ = point, D₁ = P2 _{\ E}

Remark: Nevins and Stafford: for any Artin-Schelter algebra, without affine part

• Consider (simple) modules over A₁ of gk-dimension one.

Determined by Block (1981):

B = localisation of A₁ at k[y] \ {0} (PID) The simple A₁-modules are

A/(A ∩ Bb) where b ∈ B irreducible (with technical condition on b)

and

k[x] where y acts as y − α = −_dxd , α ∈ C

Question: Is there a ’space’ parameterizing these modules?

(simple) right A₁ A₁ = H/(z − 1)

H = C_{hx, y, zi} (critical) graded right

certain objects cohP2_{q = H-modules} H-modules of gkdim 2 up to finite length where yx − xy = z2 zx = xz, zy = yz Moduli spaces modules of gkdim 1 z-torsionfree

• We work in a general setting

A = Artin-Schelter regular algebra in three variables

E smooth and O(σ) = ∞ Questions:

1. Is there a ’space’ parameterizing (critical) A-modules of gkdim 2? 2. Appearing Hilbert series?

Minimal resolutions?

3. Presentation up to lower gk-dimensional modules?

2. Some preliminaries on nc planes

• Artin-Schelter algebra of dimension 3 is (i) graded k-algebra A = k + A₁ + A₂ + . . .

global dimension 3

(ii) A has polynomial growth

(iii) A is Gorenstein, i.e. for some l ∈ Z

Exti_A(k_A, A) ∼= (

Ak(l) if i = 3,

0 otherwise. • Artin-Schelter: either 3 or 2 variables.

We consider case of 3 variables. Then l = 3 and A is Koszul, i.e.

0 → A(−3) → A(−2)3 → A(−1)3 → A → kA → 0

• They are left and right noetherian domains and

Tails(A) = GrMod(A)/ Tors(A) GrMod(A) π −→ ←− ω Tails(A)

Artin and Zhang: define projective scheme

P2_{q = Proj A := (tails(A),O, sh)}

Artin, Tate and Van den Bergh:

A −→ (E, σ, L) −→ B = B(E, σ, L) and B = A/gA where g ∈ A₃ is central

tails(A) −⊗_AB −→ ←− (−)_A tails(B) ˜ (−) −→ ←− Γ_∗ coh(E) > i∗ < i_∗

3. Moduli spaces for gk-2 modules

Describe A-modules M s.t. GKdim M = 2 Simplifications:

1. Assume that M is g-torsionfree

2. Assume that M is Cohen-Macaulay 3. Assume that M_<0 = 0, M₀ 6= 0

4. The Hilbert series of M has the form h_M(t) = e

(1 − t)2 − f

1 − t + g(t) where g(t) ∈ Z_{[t, t}−1_{]. So fix e and f .}

Note: e > 0, f ≥ 0.

A G(e, f ) cohP2_{q = Tails A} G(e,f) Moduli spaces M ∈ G(e, f ) satisfies • H0(P2_{q,M(l)) = 0 for l < 0}

• i∗M ∈ coh(E) finite dimensional length 3e Using Serre duality:

H2(P2_{q,M(l)) = Ext}2_{(O, M(l))}

∼

= Hom(M(l + 3), O)∗ = 0 for all l Using Euler form:

Via the derived equivalence of Bondal: M ∈ G(e, f ) ←→ M ∈ mod ∆ for which

• dimM = (2e + f, e + f, f ) • Hom_∆(M, p) = 0 ∀p ∈ E

• Hom_∆(p, M ) = 0 except finitely p ∈ E

In case of the Weyl algebra:

dimM = (2e + f, e + f, f ) and M (Z₀), M (Z₁) are surjective Using the relations of ∆:

Corresponds to pairs of matrices

{(X, Y ) ∈ M_2e+f(C₎2 _{| rk(Y X − XY − id) ≤ e}}

for which, up to simulaneous conjugation in Gl_2e+f(C_{), both X and Y are of the form}

f e e e + f e ∗ ∗ 0 ∗ 0 0

Still have to sort out:

• For Sklyanin algebras

• properties of these spaces

• We expect for critical modules: smooth (affine??) varieties of dimension e2 + 1

4. Hilbert series of gk-2 modules

• Sufficient: determine Hilbert series of G(e, f )inv = critical objects in G(e, f ) • Necessary conditions (Ajitabh):

If M in G(e, f )inv then h_M(t) = e (1 − t)2 − s(t) 1 − t where s(t) = P i siti ∈ Z[t] satisfies e > s₀ > s₁ > . . . ≥ 0 and P i si = f (1)

• In fact Ajitabh found necessary conditions for the appearing minimal resolutions

• s(t) satisfying (1) are represented by graphs in the form of a stair

Example:

e = 12 and s(t) = 9 + 6t + 5t2 + 4t3 + 1t6 The corresponding graph is

Counting parameters we showed that the converse is also true.

Theorem 1. There is a bijection

Hilbert functions of objects in _{G(e, f )}inv l Polynomials s ∈ Z_{[t] s.t.} e > s₀ > s₁ > . . . ≥ 0 and P i si = f given by h(t) = e (1 − t)2 − s(t) 1 − t

• Number of appearing Hilbert series is 2e−1 • We showed: Ajitabh’s necessary conditions

for minimal resulutions are also sufficient • Via Hilbert series: stratification of G(e, f )inv

there is a dimension formula for these strata unique stratum of G(e, f )inv with maximal

5. Presentation up to gk-1 modules grmod A = { f.g. right A-modules }

grmod A_≤1 = {M ∈ grmod A, GKdim M ≤ 1}

Quotient map θ : grmod A → grmod A/ grmod A_≤1 K, M ∈ grmod A are gk-1 equivalent if θ(K) ∼= θ(M )

Theorem 2 (Ajitabh and Van den Bergh). Every critical _{M ∈ G(e, f ) is gk-1 equivalent} with a critical _{K ∈ G(e, 0) s.t.}

0 → A(−1)e → Ae → K → 0

In the commutative case:

Every critical M ∈ G(e, f ) is gk-1 equivalent with a critical K ∈ G(e, e(e − 1)/2) s.t.

We proved that this is not the case for A. Example: e = 3 and

0 → A(−2)2 → A(−1) ⊕ A → M → 0 0 → A(−3) → A → K → 0

M has 10 parameters, K has 9.

Idea: only finitely many critical gk-1 modules to map M or K onto!

Corollary 1. Not every simple gk-1 over A₁ is of the form _A1/aA1, _{a ∈ A1}.

Writing A₁ = k[x, _dxd ]:

Corollary 2. Not every system of differential equations in _{x can be reduced to a single} equa-tion.